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Properties

Label 18.0.16125152370...8407.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{9}\cdot 43^{12}$
Root discriminant $32.47$
Ramified primes $7, 43$
Class number $19$
Class group $[19]$
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![968, -3412, 5544, -3175, -851, 1970, 1128, -1124, -1596, 2464, -220, -1714, 1122, 84, -304, 76, 16, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 16*x^16 + 76*x^15 - 304*x^14 + 84*x^13 + 1122*x^12 - 1714*x^11 - 220*x^10 + 2464*x^9 - 1596*x^8 - 1124*x^7 + 1128*x^6 + 1970*x^5 - 851*x^4 - 3175*x^3 + 5544*x^2 - 3412*x + 968)
 
gp: K = bnfinit(x^18 - 9*x^17 + 16*x^16 + 76*x^15 - 304*x^14 + 84*x^13 + 1122*x^12 - 1714*x^11 - 220*x^10 + 2464*x^9 - 1596*x^8 - 1124*x^7 + 1128*x^6 + 1970*x^5 - 851*x^4 - 3175*x^3 + 5544*x^2 - 3412*x + 968, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 16 x^{16} + 76 x^{15} - 304 x^{14} + 84 x^{13} + 1122 x^{12} - 1714 x^{11} - 220 x^{10} + 2464 x^{9} - 1596 x^{8} - 1124 x^{7} + 1128 x^{6} + 1970 x^{5} - 851 x^{4} - 3175 x^{3} + 5544 x^{2} - 3412 x + 968 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1612515237057830683893428407=-\,7^{9}\cdot 43^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.47$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $7, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{22} a^{11} - \frac{1}{2} a^{4} + \frac{5}{11} a$, $\frac{1}{22} a^{12} - \frac{1}{2} a^{5} + \frac{5}{11} a^{2}$, $\frac{1}{22} a^{13} - \frac{1}{2} a^{6} + \frac{5}{11} a^{3}$, $\frac{1}{16478} a^{14} - \frac{1}{2354} a^{13} + \frac{30}{8239} a^{12} - \frac{269}{16478} a^{11} + \frac{23}{749} a^{10} - \frac{3}{214} a^{9} + \frac{13}{749} a^{8} - \frac{221}{1498} a^{7} + \frac{561}{1498} a^{6} + \frac{53}{107} a^{5} + \frac{2419}{16478} a^{4} + \frac{380}{1177} a^{3} - \frac{227}{2354} a^{2} - \frac{960}{8239} a + \frac{347}{749}$, $\frac{1}{16478} a^{15} + \frac{1}{1498} a^{13} + \frac{151}{16478} a^{12} + \frac{11}{1498} a^{11} + \frac{43}{214} a^{10} - \frac{121}{1498} a^{9} - \frac{39}{1498} a^{8} + \frac{256}{749} a^{7} + \frac{25}{214} a^{6} - \frac{6359}{16478} a^{5} + \frac{75}{214} a^{4} + \frac{35}{214} a^{3} + \frac{3435}{16478} a^{2} - \frac{332}{749} a + \frac{26}{107}$, $\frac{1}{119267764} a^{16} - \frac{2}{29816941} a^{15} - \frac{193}{59633882} a^{14} + \frac{29}{1217018} a^{13} + \frac{633981}{29816941} a^{12} + \frac{252733}{29816941} a^{11} + \frac{645496}{2710631} a^{10} - \frac{48241}{2710631} a^{9} - \frac{1302017}{5421262} a^{8} - \frac{1844823}{5421262} a^{7} - \frac{214187}{59633882} a^{6} - \frac{13869150}{29816941} a^{5} - \frac{10129523}{29816941} a^{4} + \frac{11609685}{29816941} a^{3} - \frac{55065137}{119267764} a^{2} + \frac{12591827}{59633882} a - \frac{4115}{246421}$, $\frac{1}{4889978324} a^{17} + \frac{3}{1222494581} a^{16} - \frac{301}{49897738} a^{15} - \frac{21124}{1222494581} a^{14} - \frac{3358042}{1222494581} a^{13} + \frac{746269}{174642083} a^{12} + \frac{17988944}{1222494581} a^{11} + \frac{5981713}{222271742} a^{10} - \frac{304973}{20206522} a^{9} + \frac{25856987}{222271742} a^{8} + \frac{520192384}{1222494581} a^{7} - \frac{4202133}{26010523} a^{6} - \frac{463580492}{1222494581} a^{5} - \frac{355471563}{1222494581} a^{4} - \frac{7981135}{119267764} a^{3} - \frac{52085839}{349284166} a^{2} - \frac{10388200}{29816941} a - \frac{591921}{10103261}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{19}$, which has order $19$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 386705.621198 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-7}) \), 3.1.12943.1 x3, 3.3.1849.1, 6.0.1172648743.1, 6.0.634207.1 x2, 6.0.1172648743.2, 9.3.2168227525807.1 x3

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 6 sibling: 6.0.634207.1
Degree 9 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$7$7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$43$43.9.6.1$x^{9} + 1290 x^{6} + 552851 x^{3} + 79507000$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
43.9.6.1$x^{9} + 1290 x^{6} + 552851 x^{3} + 79507000$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$